1. science
2. mathematics
3. statistics

Departamento de Estadística
Universidad Carlos III de Madrid
Calle Madrid, 126
28903 Getafe (Spain)
Fax (34) 91 624-98-48
UC3M Working Papers
Statistics and Econometrics
15-10
ISSN 2387-0303
May 2015
Hierarchical Lee-Carter model estimation through data cloning
applied to demographically linked countries
Benchimol, A.a, Albarrán, I.a, Marín, J.Ma, Alonso González, P.J.b
Abstract
Some groups of countries are connected not only economically, but also social and even
demographically. This last fact can be exploited when trying to forecast the death rates of
their populations. In this paper we propose a hierarchical specification of the Lee-Carter
model and we assume that there is a common latent mortality factor for all of them. We
introduce an estimation procedure for this kind of structures by means of a data cloning
methodology. To our knowledge, this is the first time that this methodology is used in the
actuarial field. It allows approximating the maximum likelihood estimates, which are not
affected by the prior distributions assumed for the calculus. Finally, we apply the
methodology to some France, Italy, Portugal and Spain data. The forecasts obtained using
this methodology can be considered as very satisfactory.
Keywords: Bayesian inference, data cloning, hierarchical model, Lee-Carter model, longevity
risk, projected life tables.
a
b
Department of Statistics, Universidad Carlos III de Madrid.
Department of Economics, Universidad de Alcalá.
Hierarchical Lee-Carter model estimation through data cloning
applied to demographically linked countries
Benchimol, A.1 , Albarr´an, I.2 , Mar´ın, J.M.1 , and Alonso-Gonz´alez, P.J.3
(1) and (2) Statistics Department. Universidad Carlos III de Madrid.
(3) Economics Department. Universidad de Alcal´
a.
∗
Abstract
Some groups of countries are connected not only economically, but also social and even demographically. This last fact can be exploited when trying to forecast the death rates of their
populations. In this paper we propose a hierarchical specification of the Lee-Carter model and we
assume that there is a common latent mortality factor for all of them. We introduce an estimation
procedure for this kind of structures by means of a data cloning methodology. To our knowledge,
this is the first time that this methodology is used in the actuarial field. It allows approximating
the maximum likelihood estimates, which are not affected by the prior distributions assumed for the
calculus. Finally, we apply the methodology to some France, Italy, Portugal and Spain data. The
forecasts obtained using this methodology can be considered as very satisfactory.
Keywords: Bayesian inference, data cloning, hierarchical model, Lee-Carter model, longevity risk, projected life tables.
AMS subject classification: 62P05, 62F15.
1
Introduction
Insurance companies are usually facing risks of a quite different nature. Some of them have financial
roots (such as investment returns or the impact of inflation), other are related to biometric items (such as
mortality or longevity). In other cases, risks are linked to insureds and their personal situations (such as
the likely existence of disabilities), either decisions taken by them (such as surrenders, unpaid premiums
or the possibility of fraud) or due to unexpected evolution in some variables such as overheads or any
other kind of expenses.
When considering biometric risks linked to survival, it is possible to distinguish amongst three different
cases (Pitacco et al. (2009)):
(a) An individual may live a number of years around the average lifetime of his/her population. That
is, if we consider the number of annual deaths in a population at a certain age, it can be seen that
mortality fluctuates around a value with no systematic deviations across the time. This kind of
mortality is referred to individuals and it is related to random fluctuations. This phenomenon is
habitual in the life insurance business and its impact can be reduced enlarging the policies portfolio
with similar contracts. Ceteris paribus, when the portfolio is large enough, it becomes a negligible
risk but if this were not the situation, this risk could be reduced or hedged using some specific
instruments, such as reinsurance or retrocession contracts.
(b) There can be differences between recorded and expected lifetimes in a population across the years.
That is, the number of observed deaths may systematically be greater or lower than the expected.
This may be the result of a wrong mortality model specification, or a biased estimation of the
∗
Authors’
address: (1) Statistics Department, Universidad Carlos III de Madrid, C/ Madrid 126, 28903 Getafe, Spain (2)
Statistics Department, Universidad Carlos III de Madrid, Avda. Gregorio Peces-Barba 22, 28270 Colmenarejo, Spain. (3)
Economics Department, Universidad de Alcal´
a, Pl. Victoria 2,28802, Alcal´
a de Henares, Spain. E–mails: A. Benchimol,
[email protected], I. Albarr´
an, [email protected], J.M. Mar´ın [email protected],
P.J. Alonso-Gonz´
alez, [email protected]
1
relevant parameters of the model. An example of this situation is the longevity risk, situation in
which the insureds live more years than the expected. Unlike the former case of mortality, this
one is concerned with the whole population. So, the solution of increasing the portfolio size has no
effect in hedging against this risk. Some authors (see Cox et al. (2010), Blake et al. (2006) and
Lin and Cox (2005)) suggest the use of specific financial instruments, such as mortality derivatives,
to protect insurance companies against the economic effects of these contingencies.
(c) It is possible that death rates may suffer a sudden increase due to the presence of some critical life
conditions, such as epidemics, natural disasters, severe weather conditions or wars. This is called
catastrophic risk and it has a strong impact in the short term. As longevity risk, it is linked to
the whole population, but the scope in time is quite different because longevity risk is referred
to the medium and long term. Regardless the nature of the disaster, the companies require an
appropriate protection against the effects of them. Increasing the size of the insured portfolio is
not an effective measure to hedge against these contingencies. The right decision is diversification.
Due to its non-individual scope, it can be obtained through risk transfer agreements. This solution
is another difference between this risk and that related with longevity because this last one affects
the whole portfolio.
The first kind of risk is linked to random fluctuations around a certain value whereas the last one is
concerned with the effects of unexpected events. Given a specific environmental conditions, both of them
can not be predicted. However, the second situation is referred to errors in the life tables estimation
process that can be improved. These tables are frequently used by actuaries in their professional activity
in life insurance business. As they can be used in a great number of tasks (such as calculating premiums,
annuities, pension benefits or reserves), it is paramount to be sure that they are rightly calibrated and
that the estimated probabilities reflected in them do not include any possible bias. If it is the case, the
consequences will be passed to the financial statements resulting in lesser profits or even bankruptcy.
A good life tables generator should produce sets of probabilities with the ability of incorporating the
decreasing trend in death probabilities at every age. That is, it would be desirable that increasing life
expectancies could be deduced from them. Although the enlargement of the life span has been one
of the most relevant achievement made by the mankind during the last century, it is also one of the
most disturbing financial challenges for the next decades. The insurance companies have been forced
to allocate more capital to sustain their existing annuity business to prevent from adverse effects on
reserves and profitability, and therefore to protect the solvency of companies. These effects are related to
the fact that the longer the life is, the greater the number of years in retirement. Of course, everybody
wants to live this extra time as good as possible. However, an increasing amount of funds is needed
to get this goal. This is the root of the problem: neither public nor private pensions schemes are
nowadays able to attend these financial requirements. Public Social Security systems are facing with the
consequences of the population ageing process, whereas the private systems must take into account that
the amounts reserved for those future payments were calculated in the past using previous life tables.
The life expectancies derived of those tables are systematically lesser than those really recorded. This
fact supposes not only a negative effect in the liquidity but also into the solvency of the undertakings
devoted to this business. This situation is called longevity risk. As some authors suggest, this risk arises
when a population enjoys of survival time greater than expected (Cairns et al. (2006)).
Reactions to the problem of longevity risk are mainly two. On one hand, actuaries have been trying
to create better models to reflect the improvements in mortality, paying more attention to the levels of
uncertainty of predictions (see Lee and Carter (1992), Cairns et al. (2006) and Pedroza (2006)). On
the other hand, actuaries are trying to do the same as Capital Markets do to share risks, through the
development of financial instruments linked to mortality (see Cox et al. (2010). Blake et al. (2006) and
Lin and Cox (2005)). This paper is devoted to introduce a model able to capture these improvements.
The final motivation obeys to actuarial purposes: to protect insurance companies in respect of mortality
improvements. Actuaries in insurance companies must rely on tables that include future trends in
mortality. This is known as projected and dynamic life tables.
As it has been pointed out before, mortality has been reducing across years during the last decades,
sometimes faster, sometimes slower. So, this process can be treated from a statistical point of view and
the evolution of this variable can be reflected using a stochastic model. We can distinguish between two
different kinds of models: classical vs. Bayesian. The first group uses techniques based on more or less
complex regressions. They use sample information without including any prior information. Within this
group, Lee-Carter model (Lee and Carter (1992)) is one of the most popular among users. It considers a
relation between death rates independent of years and a mortality index that is variable across the time
2
and independent of ages. An extension of this model is due to Renshaw and Haberman (2006). They
included an extra variable that tries to reflect the cohort effect in mortality. More complex models are
those based on the use of smoothing techniques. The first one was introduced by Currie et al. (2004).
They suggested the use of B-splines and P-splines for fitting mortality surfaces. Afterwards, the APC
model (stands for Age-Period-Cohort) was introduced by Currie et al. (2006), that can be considered as
a particular case of the Renshaw-Haberman model. The difference between both models is that the APC
was estimated using P-splines in order to assure a smooth result. Finally, models including two or more
factors are a subset within the classical group. The seminal work was the CBD model (stands for Cairns,
Blake and Dowd) due to Cairns et al. (2006). Its main assumption is that the effects linked to age and
period are different between them and they both affect the future death rate. Further development of
this basic model can be found in Cairns et al. (2009) and in Dowd et al. (2010). They include cohort
effect and quadratic terms.
Relevant information based on historical information or in skilled opinions are used in Bayesian
models to improve the estimation. From an actuarial point of view, a projected life table can be used
as a starting point to achieve better estimations of the future mortality. Pedroza (2006) is a good
example of this methodology. She proposed a Bayesian approach to the Lee-Carter model considering
the uncertainty in the age parameters, as well as in the usually forecast mortality index. Expert-based
probabilistic population projections have been produced by Lutz et al. (1998), Lutz et al. (2004) and
Lutz et al. (2008). However, this method is not explicitly based on available data, and instead relies on
a collection of experts and their ability to specify probabilistic bounds, that may or may not be accurate
(Alho (2005)). Besides, Dellaportas et al. (2011) used a Bayesian version of the Heligman and Pollard
(1980) to predict mortality figures.
As it has been pointed out before, econometric models are one of the most used statistical tools to
analyse and project death rates. General Linear Models (GLM) can be applied assuming that age, cohort
and/or period are explanatory variables. Estimates can be obtained after using maximum likelihood
techniques which can be difficult to obtain in complex models. An alternative estimation methodology is
the data cloning technique that attempts to estimate a GLM model but using a MCMC based algorithm.
The two seminal papers about this methodology are Lele et al. (2007) and Lele et al. (2010), with
applications in complex ecological models. After that, the technique has been applied in other fields like
Finance (see Mar´ın et al. (2015)). The algorithm is based on the technique called simulated annealing
(Brooks and Morgan (1995)). This is a computationally intensive method, but the algorithm can be
easily implemented with the R library dclone (S´olymos (2009)).
This paper deals with the estimation of a hierarchical model through data cloning. Our goal is to
estimate a global Lee-Carter model (Lee and Carter (1992)) for a group of countries that are supposed
to be close related. Therefore, the corresponding parameters must be linked among them. In such case,
we can assign prior distributions that are related by means of a given set of hyperprior distributions with
some specific parameters (hyperparameters). As far as our knowledge reaches, this is the first paper that
tries to use this methodology in the actuarial field.
The remainder of this paper is organized as follows. Section 2 gives an overview of the Lee-Carter
model, its fundamentals and some developments derived from it. Section 3 shows the foundations of the
Bayesian approach to the former model. Section 4 describes and introduces the data cloning methodology.
Section 5 describes the validation of the estimation procedure followed in the paper. Section 6 reflects
the use of this algorithm to real data. Finally, Section 7 summarizes the main conclusions and final
remarks.
2
Lee-Carter Model (LC )
Lee and Carter (1992) proposed a model to forecast mortality as a function of a time-varying index.
This paper was the seminal work for further developments in the estimation of future mortality, such as
Lee (2000), Booth et al. (2002), Li and Lee (2005) and Pedroza (2006).
The LC model deals with mx (t), the central death rate for age x in year t (it is calculated as the
ratio of deaths to mid-year population size for a given interval of age and time). The LC model can be
expressed as:
log [mx (t)] = αx + βx κt + εx.t .
(1)
where αx parameters describe the pattern of the average mortality at each age, while the parameters βx
describe deviations from this average pattern when κt varies. Both set of parameters, {αx } and {βx }
are independent from time. The variable, κt , is an index. It can be expressed as a time series process
3
and describes the change in the level of mortality over time. It is important to notice that this index
is an unobservable variable, so it must be estimated. Finally, εx.t , the error term, denotes the random
deviations unexplained by the model, and it is assumed to have mean 0 and constant variance σε2 .
The expression βx κt indicates that it is assumed that there is no interaction between age and time
variables: βx parameters are fixed over time and κt parameters are fixed throughout ages. This means
that βx will be the same during all the years and κt will be the same for all ages. The LC model is quite
useful in cases when the only available information is that referred to central death rates, mx (t). These
figures are easily get from census data.
When a frequentist approach is used, the parameters can be estimated by least squares using the first
element of the singular value decomposition of the following matrix:


log [m
b x1 (t1 )] − α
bx1 · · · log [m
b x1 (tn )] − α
b x1


..
..
..
(2)
Z=
.
.
.
.
log [m
b xm (t1 )] − α
b xm
···
log [m
b xm (tn )] − α
b xm
Since the system of equations of the LC model is underdetermined, two additional constraints on the
parameters βx and κt have to be imposed. The indeterminacy of the LC model is due to the likelihood
associated with the model, which has an infinite number of equivalentP
maxima, each of
Pwhich produces
the same estimations. Lee and Carter (1992) proposed as constraints x βx = 1 and t κt = 0. Under
this set of constraints, αx is the average value of the logarithm of the central death rate for age x in the
considered sample, whereas βx is the percent change of the natural logarithm of the central death rate
at a certain age due to changes in the mortality index in a certain year.
Once the values of αx , βx and κt have been estimated, it is needed a pattern for the future evolution
of the mortality index. LC model supposes that κt can be modelled as a random walk with drift:
κt = κt−1 + θ + ωt .
where θ is the drift parameter which models a linear trend and ωt is an error term. Thus, the forecast
values of the unobservable process κt for r periods after the last observation at time n is calculated as
follows:
b
κ
bn+r = κ
bn + rθ.
Finally, the forecast death rates at age x for year n + r can be obtained including this last estimation
in the equation. The results for m
b x (n + r) can be expressed as:
log(m
b x (n + r)) = α
bx + βbx κ
bn+r
The variance of the forecast mortality index κ
bn+r r periods ahead can be expressed as (see Pedroza
2006):
σ
bκ2 n+r = r2 σ
bθ2 + rb
σω2
where σ
bθ2 is the estimate of the variance of θ and σ
bω2 is the estimate of the variance of the error term ω.
Thus, the LC (1 − α) × 100% prediction intervals for the log-central death rates are calculated as follows:
bκn+r
log(m
b x (n + r)) ± z α2 σ
3
Bayesian approach to the LC model
This section shows a Bayesian statistical methodology for making inferences about the parameters of a
LC model for several related populations by means of a hierarchical Bayesian model. Originally, the
Bayesian approach of the LC model applied to one population was studied by Pedroza (2006) who
analysed U.S. mortality data.
Hierarchical modelling is used when information about observational units is available on different
levels. The hierarchical form of analysis and organization is quite useful regarding multi-parameter
problems (see Gelman et al. (2014)). This paper is concerned to a case that considers several populations
with some social-economic characteristics in common. For this reason, the parameters of the LC model
can be treated as connected in some way, implying that the dependence among them can be reflected
with a joint probability model.
4
(j)
Let be j = 1. . . . , J populations, in such a way that mx (t) is the central death rate for an age x at
time t and population j, such that the corresponding LC model can be expressed as:
h
i
(j)
log m(j)
= αx(j) + βx(j) κt + εx.t
x (t)
κt
= κt−1 + θ + ωt .
where
∼ N 0; σε2(j) .
∼ N 0; σω2 .
(j)
εx.t
ωt
(j)
As usual, we observe mx (t) whereas κt is unobserved. The goal is to estimate the set of parameters
(j)
(j)
αx , βx and κt and use them to forecast the central death rates at each age for each generation in each
population.
We assume proper prior distributions with a hierarchical structure. We consider normal distributions
(j)
(j)
for αx parameters and Dirichlet distributions for βx ones. in order to include the restriction about
P
x βx = 1, we assume inverse-gamma distributions for variances as it is a conjugate distribution in a
normal model, allowing to deal with most of real cases about prior information regarding the parameters
of the model.
In particular, we assume as prior distributions for the parameters (for j = 1. . . . .J):
αx(j)
βx(j)
σε2(j)
κ1
θ
2
σω
∼
∼
∼
∼
∼
∼
2(j)
N (µ(j)
x , σx ).
Dirichlet(1, 1, . . . , 1).
InvGamma(γ1 , γ2 ).
N (0, σω2 ).
N (0, 100).
InvGamma(1, 1).
(3)
And the hyperprior distributions of the hyperparameters are
µ(j)
x
2(j)
σx
γ1
γ2
∼
∼
∼
∼
N (0, 10).
InvGamma(1, 1).
Gamma(1, 1).
Gamma(1, 1).
By other hand, the likelihood function for this model is:

h
i

2 
(j)
(j)
(j)
J Y
n Y
ω
Y
κ
log
m
(t)
−
α
−
β
x
x
x
t
1
 1
 
exp − 
L m(j)
.
√
x (t); Θ =
(j)
(j)
2
2πσ
σ
ε
ε
j=1 t=1 x=1
(j)
(j)
(j)
2(j)
2(j)
where Θ = αx , βx , κ1 , µx , σx , σε , θ, σω2 , γ1 , γ2 for j = 1, . . . , J and x = 1, . . . , ω.
The joint posterior distribution for all the parameters is obtained by multiplying the likelihood
function by the corresponding prior distributions (3). In general, the full set of conditional distributions
is required to implement a MCMC algorithm. Then, the conditional posterior distributions of each
parameter is easily obtained from the joint posterior distribution, considering only the proportional
terms to each parameter.
4
The data cloning methodology
The data cloning method is a simulation technique to compute maximum likelihood estimates of parameters along with their asymptotic variances, by using a MCMC methodology (see Lele et al. (2007) and
Lele et al. (2010)). It uses the simplicity of Monte Carlo algorithms to calculate maximum likelihood
estimations in models whose complexity make necessary the use of high-dimensional integration to obtain
them. The methodology is based on the basic idea of repeating an experiment several times conditioned
to obtain always the same data.
5
Let us denote log [mx (t)] as yt for (t = 1, . . . , n) in such a way that y = (y1 , . . . , yn ). In a MCMC
procedure, once data y has been observed, the samples from the posterior distribution π(Θ|y) are
generated. This posterior distribution is proportional to the product of the likelihood function L(Θ|y)
and a given proper prior distribution π(Θ). Then, in data cloning, samples are generated from the
K
posterior distribution, π (K) (Θ|y), that is proportional to the K th power of the likelihood, [L(Θ|y)] ,
multiplied by a proper prior distribution, π(Θ).
K
The expression [L(Θ|y)] is the likelihood for K copies of the original data and, for K large enough,
π (K) (Θ|y) converges to a multivariate normal distribution with mean equal to the maximum likelihood
estimates of the parameters, and covariance matrix equal to 1/K times the inverse of the Fisher information matrix for the maximum likelihood estimates (see Lele et al. (2007)).
Once the samples have been obtained with a MCMC procedure, sample means are computed based on
the posterior distributions of the parameters. They provide an approximation of the maximum likelihood
estimates of the parameters.
As a summary, the data cloning algorithm follows the next steps:
Step 1: Create K-cloned data set y(K) = (y, y, . . . , y), where the observed data vector is repeated K
times.
Step 2: Using an MCMC algorithm, generate random numbers from the posterior distribution that is
based on a prior π(Θ) and the cloned data vector y(K) = (y, y, . . . , y), where the K copies of y are
assumed to be independent of each other. In practice, any proper prior distribution can be used.
Step 3: Compute the sample mean and variances of each of the individual values of the vector of
parameters Θ (for M iterations of the MCMC algorithm) generated from the marginal posterior
distribution. The maximum likelihood estimates of Θ and the approximate variances of the maximum likelihood estimates are those referred to the posterior mean values and to K times the
posterior variances. respectively.
The algorithm has been programmed using the package dclone (S´olymos (2009)) from the R project
(R Core Team. R Foundation for Statistical Computing (2012)). The optimal number of clones has been
established considering some statistics computed in the package dclone (S´olymos (2009)), such as the
maximum eigenvalue of the posterior covariance matrix, the minimum squared error and the squared
error (see Lele et al. (2010)).
5
Validating the estimation procedure
Before applying the procedure to real data, we check it by means of a validation study. The validation
has been done by simulating an array of log-central death rates, assuming 4 populations, 41 consecutive
ages and 50 consecutive calendar years each.
In order to simulate data we took as a starting point the estimated parameters for some French
data taken from the Human Mortality Database (see www.mortality.org) for ages between 60 and 100,
and for calendar years between 1960 and 2009, including both ends in both cases. The estimation was
undertaken by the standard frequentist procedure based on the singular value decomposition raised by
Lee and Carter (1992) using R .
(1)
For population 1, we generated each parameter αx as the respective French α
ˆ x parameter plus a
(j)
random component following a N(µ=0;σ=0.001) distribution. For populations 2, 3 and 4, αx parameters
were generated taking 95%, 90%, and 105% of French α
ˆ x parameters, plus a random component following
(j)
N(µ=0;σ=0.002). N(µ=0;σ=0.003) and N(µ=0;σ=0.004) distributions, respectively. Regarding βx
parameters for each population, they were simulated by four vectors of dimension 41, each of them
following
a Dirichlet distribution with all its parameters equal to 1, in order to fulfill the constraint
P
β
=
1.
x x
On the other side, regarding the mortality index κt , adjusting for France estimates, parameters were
assumed to be as κ
ˆ 0 = −11.2075, θˆ = −0.5728 and σ
ˆω2 = 5. Thus, κt was fixed as a vector of 50 values
from t = 1960 up to t = 2009, where each κt was a random value generated from a normal distribution
with mean κt−1 + θˆ and variance σ
ˆω2 .
h
i
(j)
Finally, the data set of log-central death rates log mx (t) (four matrices of dimensions 50 × 41, one
(j)
(j)
for each population) was simulated by means of four normal distributions with means αx + βx κt and
2(1)
2(2)
2(3)
2(4)
variances σy =0.0010, σy = 0.0015, σy = 0.0020 and σy =0.0025, respectively.
6
We used the package dclone (S´
olymos (2009)) from the R project (R Core Team (2015)) in order to
program the algorithm. We checked the optimal number of clones by means of some statistics computed
in the package dclone (S´
olymos (2009)), such as the maximum eigenvalue of the posterior variance,
the minimum squared error and the squared error (see Lele et al. (2010)). There were not relevant
improvements in their values when the number of clones was larger than 5, and therefore we worked with
5 clones to estimate the parameters. Besides, we used the same prior distributions as in Section 3.
Then, the parameters of this hierarchical model were estimated applying our algorithm with the data
cloning technique, on the same simulated data set 100 times, generating thus 100 replicates, and the
results were assessed.
Let us consider the Pearson’s coefficient of variation (CV
q ) and the the relative mean squared error
¯
ˆ
ˆ
ˆ
(RMSE ) respectively as CV (θ) = σ ˆ/|θ| and RM SE(θ) = E[(θˆ − θ)2 ]/|θ|.
θ
We calculated both relative dispersion measures for hthe 8200
i (50 calendar years × 41 ages × 4
(j)
populations) estimators of the log-central death rates, log m
b x (t) . The mean CV for the whole sample
was 0.048, and the mean RMSE was 0.047. In the case of the CV, 154 out of 8200 were larger than 0.2.
while in the case of the RMSE, 171 out of 8200 were larger than this threshold (1.87% and 2.08% of the
sample, respectively). These results suggest that the procedure seems to be unbiased and stable.
In Figures 1a and 1b the respective histograms of both measures for the whole sample of 8200
estimators are shown.
FIGURES 1a and 1b ROUND HERE
6
Application to real data
The data cloning methodology has been applied to a set of European countries mortality data. We have
taken the central death rates from France, Italy, Portugal and Spain located in the Human Mortality
Database (see www.mortality.org).
We have focused a time span between years 1960 and 2009 and ages between 60 and 100 years old.
These countries present a similar social development and their populations enjoy parallel welfare states.
More specifically, they show similar standard demographic indices such as:
– Life expectancy at birth (LEB): it is the average number of years that a newborn can expect to
live, according to the mortality conditions at his/her birth time.
– Life expectancy at age 65 (LE65): it is the average number of years that a person age 65 can expect
to live, according to the mortality conditions at the time he/she attains that age.
– Median age of the population (MAP): it is the median of ages of the alive people in a country at
a certain time. It shows the progressive ageing in the population as a result of the low birth and
death rates, yielding in a high life expectancy. This index is also affected by the immigration flows.
– Old-dependency ratio (ODR): it is the proportion between the number of people over 65 years old
and the number of people between 16 and 64 years old. Namely, it is the ratio between retired and
active people (the last one is the sum of employed and unemployed workers). ODR index is used
to measure the pressure of the elderly people over the productive population.
The corresponding values of these indices for years 2012-2013 for each country are shown in table 1.
TABLE 1 ROUND HERE
In order to validate the predictive performance of the model, the data set was split into two groups:
the first one (training sample) includes data from 1960 to 1999, whereas the second one (validation
sample) includes data from 2000 to 2009.
We complete the analysis of the hierarchical model by applying the data cloning technique. We
have programmed the algorithm using package dclone (S´olymos (2009)) from the R project (R Core
Team (2015)). We check what is the optimal number of clones, regarding some statistics computed
in the package dclone (S´
olymos (2009)), such as the maximum eigenvalue of the posterior variance,
the minimum squared error and the squared error (see Lele et al. (2010)). There are not relevant
improvements in their values when the number of clones is larger than 5, therefore we use this number
of clones to analyse the data. We have used the same prior distributions as in Section 3.
7
Mean and standard deviation of parameters were estimated for the training sample using 5 clones after
(j)
(j)
50,000 iterations of the MCMC algorithm. The estimated mean for the {αx } and {βx } parameters for
each country are shown in table 2.
TABLE 2 ROUND HERE
2(j)
On the other hand, error variances σx
estimated for each country are shown in table 3.
TABLE 3 ROUND HERE
(j)
(j)
Using the estimations of αx . β
age x and each country, along with the forecasted values
h x for each
i
(j)
κt , the log-central death rates log mx (t) were estimated for the period 2000-2009.
The 95% prediction intervals for the predicted values, based on the Wald approximation, and the
actual values of the log-central death rates are shown in tables 4a, 4b, 4c and 4d for France, Italy,
Portugal and Spain, respectively, for ages x = 60, 70, 80, 90 and 100, and for a prediction horizon
t = 2000, . . . , 2009. Notice that all intervals include the actual values of the parameters.
TABLES 4a, 4b, 4c and 4d ROUND HERE
In Figures 2a, 2b, 2c and 2d, we represent the mortality surfaces for France, Italy, Portugal and
(j)
Spain, respectively. Here, we show first the observed central death rates mx (t) (height of the surface)
for t = 1960, . . . , 2009. And then to the right, the projected central death rates for t = 2000, . . ., 2009
forecasted by our model with the data in the training sample. As it can be seen, the projected surfaces
suggest a proper fit.
FIGURES 2a, 2b, 2c and 2d ROUND HERE
7
Concluding remarks
We have introduced a hierarchical Lee-Carter model to forecast the death rates of a set of demographically
linked countries. Although it has been assumed that each country has its own specific characteristics,
the model is based on the existence of a common and latent mortality structure. This idea is quite
interesting to estimate the parameters of the model because it allows to take advantage of the whole
set of information, that is, the forecasts of a certain country are calculated not only based on its death
rates but also in those of the rest of the considered linked populations. Bayesian methodology is a very
effective way to deal with hierarchical models. However, this scheme is limited by the fact that it is often
necessary that the analyst determines the prior distributions for all parameters and hyperparameters of
the model. Data cloning is an alternative to surpass the previous limitation and it allows to approximate
the maximum likelihood estimates. In this scenario, the role of the prior distributions is not determinant.
Although this methodology has been applied to areas such as Ecology and recently in Finance, it is
the first time that data cloning has been used in the Actuarial field, to our knowledge extent. The set of
information includes the central death rates of France, Italy, Portugal and Spain. In order to check the
validity of the forecasts, the sample has been divided into two sets. The first one is devoted to estimate
the parameters, whereas the second one is used to contrast the accuracy of the results. The model is
able to rightly predict the central death rates rates in all cases, using 95% approximated prediction
intervals. All of these results can be directly used in the management of private and/or public pension
systems, as nowadays one of the most relevant problems for the insurance industry is connected with
wrong estimations of survival probabilities.
Future research will involve the implementation of this methodology with other kind of models used
to forecast death rates, such as Cairns-Blake-Dowd (CBD) stochastic mortality models, or those based
on P-splines.
References
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10
LEB
Male
Female
France
78.7
85.4
Italy
79.8
84.8
Portugal
77.3
83.6
Spain
79.5
85.8
Source: Eurostat 2012, 2013.
LE65
Male
Female
19.1
23.4
18.5
22.1
17.6
21.3
18.7
22.8
MAP
Male
Female
38.2
41.2
43.0
45.6
37.6
41.9
40.1
42.9
ODR
General
27.5
32.7
29.4
26.3
Table 1: Demographic indices
x
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
France
αx
βx
-4.2041
0.0308
-4.1258
0.0311
-4.0550
0.0326
-3.9676
0.0319
-3.8957
0.0325
-3.7844
0.0294
-3.7493
0.0343
-3.6656
0.0342
-3.5808
0.0343
-3.5056
0.0347
-3.4138
0.0347
-3.3321
0.0349
-3.2398
0.0338
-3.1532
0.0344
-3.0647
0.0346
-2.9568
0.0331
-2.8762
0.0342
-2.7751
0.0333
-2.6717
0.0322
-2.5771
0.0324
-2.4660
0.0304
-2.3543
0.0285
-2.2371
0.0263
-2.1387
0.0253
-2.0372
0.0242
-1.8042
0.0001
-1.8385
0.0230
-1.7166
0.0183
-1.6406
0.0194
-1.5488
0.0186
-1.4528
0.0174
-1.3577
0.0171
-1.2756
0.0150
-1.1823
0.0125
-1.0953
0.0106
-1.0279
0.0105
-0.9569
0.0099
-0.8869
0.0091
-0.8193
0.0081
-0.7537
0.0070
-0.6875
0.0055
Italy
αx
βx
-4.3192
0.0354
-4.2155
0.0344
-4.1221
0.0340
-4.0354
0.0338
-3.9293
0.0320
-3.8367
0.0320
-3.7390
0.0306
-3.5864
0.0232
-3.5521
0.0291
-3.4624
0.0295
-3.3750
0.0299
-3.2760
0.0279
-3.1840
0.0290
-3.0938
0.0286
-2.9978
0.0282
-2.9048
0.0280
-2.8027
0.0267
-2.7063
0.0263
-2.6167
0.0269
-2.5120
0.0256
-2.4158
0.0273
-2.3168
0.0260
-2.2226
0.0257
-2.1167
0.0247
-2.0232
0.0240
-1.9248
0.0231
-1.8302
0.0224
-1.7336
0.0218
-1.6409
0.0210
-1.5542
0.0210
-1.4605
0.0204
-1.3760
0.0190
-1.2849
0.0187
-1.2042
0.0168
-1.1280
0.0171
-1.0624
0.0160
-0.9878
0.0149
-0.8986
0.0117
-0.8515
0.0131
-0.7912
0.0127
-0.7292
0.0116
Portugal
αx
βx
-4.2101
0.0292
-4.1261
0.0262
-4.0465
0.0304
-3.9441
0.0296
-3.8708
0.0302
-3.7692
0.0292
-3.6857
0.0293
-3.5832
0.0294
-3.4903
0.0311
-3.3903
0.0292
-3.2894
0.0332
-3.1739
0.0267
-3.0996
0.0324
-2.9929
0.0310
-2.8906
0.0316
-2.7885
0.0317
-2.6748
0.0301
-2.5766
0.0303
-2.4658
0.0307
-2.3615
0.0287
-2.3062
0.0269
-2.2115
0.0246
-2.1235
0.0271
-2.0080
0.0243
-1.9233
0.0250
-1.8301
0.0248
-1.7338
0.0227
-1.6204
0.0197
-1.5465
0.0215
-1.4672
0.0219
-1.3737
0.0189
-1.3090
0.0182
-1.1934
0.0152
-1.1482
0.0173
-1.0862
0.0175
-0.9946
0.0148
-0.9286
0.0143
-0.8617
0.0133
-0.7910
0.0115
-0.7387
0.0116
-0.6632
0.0085
(j)
(j)
Spain
αx
βx
-4.3799
0.0302
-4.3003
0.0258
-4.2187
0.0331
-4.1232
0.0311
-4.0293
0.0324
-3.9388
0.0332
-3.8473
0.0308
-3.7519
0.0321
-3.6611
0.0333
-3.5676
0.0305
-3.4699
0.0345
-3.3762
0.0286
-3.2755
0.0367
-3.1636
0.0322
-3.0673
0.0335
-2.9760
0.0327
-2.8400
0.0287
-2.7669
0.0300
-2.6723
0.0339
-2.5625
0.0278
-2.4695
0.0308
-2.3669
0.0241
-2.2670
0.0298
-2.1734
0.0260
-2.0716
0.0291
-1.9815
0.0251
-1.8778
0.0244
-1.7799
0.0222
-1.6890
0.0233
-1.5916
0.0191
-1.5028
0.0193
-1.3907
0.0033
-1.3331
0.0138
-1.2694
0.0116
-1.1708
0.0118
-1.1056
0.0117
-1.0321
0.0106
-0.9621
0.0096
-0.8964
0.0088
-0.8278
0.0073
-0.7704
0.0070
Table 2: Point estimates of αx and βx parameters
11
Parameter
σε2(F rance)
σε2(Italy)
σε2(P ortugal)
σε2(Spain)
Point Estimate
0.0022
0.0023
0.0025
0.0014
Table 3: Error variances estimates
(a) France
t
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
t
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
95%
Up B.
-4.7128
-4.8500
-4.9636
-5.0685
-5.1645
-5.2567
-5.3435
-5.4298
-5.5138
-5.5926
x=60
PI
Actual value
Low B.
-3.9458
-4.4295
-3.8354
-4.5117
-3.7463
-4.4671
-3.6671
-4.4559
-3.5965
-4.5236
-3.5288
-4.5153
-3.4674
-4.5394
-3.4073
-4.5413
-3.3487
-4.5631
-3.2943
-4.5516
95%
Up B.
-3.9740
-4.1305
-4.2609
-4.3786
-4.4888
-4.5919
-4.6924
-4.7886
-4.8817
-4.9730
x=70
PI
Actual value
Low B.
-3.1368
-3.6005
-3.0090
-3.6439
-2.9074
-3.6412
-2.8188
-3.6893
-2.7362
-3.7474
-2.6616
-3.7872
-2.5902
-3.8231
-2.5226
-3.8467
-2.4572
-3.8839
-2.3955
-3.9001
95%
Up B.
-1.8009
-1.8655
-1.9237
-1.9790
-2.0279
-2.0791
-2.1249
-2.1721
-2.2179
-2.2609
x=90
PI
Actual value
Low B.
-1.2453
-1.5477
-1.1954
-1.5622
-1.1512
-1.5732
-1.1110
-1.5299
-1.0758
-1.6495
-1.0391
-1.5507
-1.0080
-1.7740
-0.9752
-1.6663
-0.9427
-1.6750
-0.9147
-1.7800
95%
Up B.
-0.9304
-0.9413
-0.9512
-0.9617
-0.9730
-0.9839
-0.9956
-1.0048
-1.0183
-1.0291
x=100
PI
Actual value
Low B.
-0.4895
-0.7227
-0.4828
-0.7344
-0.4780
-0.7267
-0.4719
-0.6907
-0.4645
-0.7840
-0.4583
-0.7511
-0.4512
-0.7869
-0.4465
-0.7803
-0.4371
-0.7791
-0.4317
-0.7737
95%
Up B.
-4.8909
-5.0499
-5.1816
-5.3034
-5.4134
-5.5203
-5.6238
-5.7188
-5.8175
-5.9069
x=60
PI
Actual value
Low B.
-4.0362
-4.6628
-3.9076
-4.6356
-3.8045
-4.7128
-3.7119
-4.6450
-3.6305
-4.7028
-3.5526
-4.7689
-3.4781
-4.7879
-3.4124
-4.7855
-3.3427
-4.8258
-3.2828
-4.8757
95%
Up B.
-3.8806
-4.0097
-4.1192
-4.2206
-4.3126
-4.4024
-4.4877
-4.5708
-4.6518
-4.7274
x=70
PI
Actual value
Low B.
-3.1145
-3.5972
-3.0106
-3.6276
-2.9252
-3.6861
-2.8484
-3.7132
-2.7804
-3.7491
-2.7153
-3.7828
-2.6554
-3.8509
-2.5979
-3.8849
-2.5407
-3.8709
-2.4889
-3.9394
95%
Up B.
-1.8463
-1.9261
-1.9971
-2.0629
-2.1228
-2.1829
-2.2381
-2.2939
-2.3487
-2.3990
x=90
PI
Actual value
Low B.
-1.2411
-1.5906
-1.1781
-1.5580
-1.1236
-1.5955
-1.0755
-1.5318
-1.0319
-1.6455
-0.9880
-1.6189
-0.9500
-1.6573
-0.9112
-1.6724
-0.8733
-1.6676
-0.8390
-1.7730
95%
Up B.
-1.0218
-1.0580
-1.0905
-1.1223
-1.1541
-1.1834
-1.2137
-1.2431
-1.2702
-1.2988
x=100
PI
Actual value
Low B.
-0.5293
-0.7425
-0.5028
-0.7473
-0.4797
-0.7606
-0.4577
-0.7106
-0.4348
-0.8115
-0.4151
-0.7651
-0.3945
-0.8031
-0.3753
-0.7857
-0.3571
-0.7717
-0.3380
-0.7741
95%
Up B.
-2.9710
-3.1051
-3.2169
-3.3199
-3.4149
-3.5042
-3.5915
-3.6755
-3.7573
-3.8348
x=80
PI
Actual value
Low B.
-2.2088
-2.5806
-2.1003
-2.6520
-2.0137
-2.6533
-1.9357
-2.6526
-1.8647
-2.7465
-1.8004
-2.7589
-1.7383
-2.7926
-1.6796
-2.8166
-1.6232
-2.8341
-1.5706
-2.8810
95%
Up B.
-2.8838
-3.0013
-3.0990
-3.1909
-3.2751
-3.3551
-3.4315
-3.5070
-3.5804
-3.6488
x=80
PI
Actual value
Low B.
-2.1691
-2.4969
-2.0753
-2.6571
-1.9990
-2.6116
-1.9292
-2.5919
-1.8678
-2.6863
-1.8101
-2.6757
-1.7571
-2.7243
-1.7033
-2.7186
-1.6518
-2.7292
-1.6054
-2.7952
(b) Italy
t
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
t
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
12
(c) Portugal
t
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
t
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
95%
Up B.
-4.7066
-4.8325
-4.9381
-5.0367
-5.1270
-5.2140
-5.2962
-5.3785
-5.4562
-5.5290
x=60
PI
Actual value
Low B.
-3.9522
-4.4006
-3.8508
-4.4030
-3.7692
-4.5044
-3.6946
-4.4533
-3.6272
-4.5181
-3.5650
-4.4892
-3.5075
-4.4954
-3.4490
-4.4733
-3.3954
-4.5403
-3.3465
-4.5708
95%
Up B.
-3.8358
-3.9827
-4.1050
-4.2187
-4.3214
-4.4198
-4.5145
-4.6057
-4.6952
-4.7800
x=70
PI
Actual value
Low B.
-3.0136
-3.4477
-2.8943
-3.4950
-2.7993
-3.5153
-2.7128
-3.4940
-2.6370
-3.6001
-2.5651
-3.6227
-2.4988
-3.6885
-2.4354
-3.7206
-2.3725
-3.7285
-2.3146
-3.7243
95%
Up B.
-1.7562
-1.8262
-1.8884
-1.9470
-2.0037
-2.0542
-2.1069
-2.1592
-2.2075
-2.2542
x=90
PI
Actual value
Low B.
-1.1486
-1.4443
-1.0944
-1.4653
-1.0482
-1.4877
-1.0049
-1.4203
-0.9639
-1.5946
-0.9286
-1.4342
-0.8914
-1.5836
-0.8559
-1.5311
-0.8224
-1.4946
-0.7905
-1.6661
95%
Up B.
-0.9781
-0.9989
-1.0189
-1.0408
-1.0614
-1.0813
-1.1012
-1.1219
-1.1418
-1.1615
x=100
PI
Actual value
Low B.
-0.4296
-0.6758
-0.4157
-0.6769
-0.4023
-0.6744
-0.3875
-0.6642
-0.3738
-0.7622
-0.3609
-0.6674
-0.3485
-0.7414
-0.3347
-0.7582
-0.3217
-0.7629
-0.3099
-0.7741
95%
Up B.
-4.8579
-4.9948
-5.1104
-5.2139
-5.3089
-5.4001
-5.4855
-5.5687
-5.6523
-5.7290
x=60
PI
Actual value
Low B.
-4.1486
-4.5375
-4.0364
-4.5612
-3.9457
-4.5602
-3.8671
-4.5422
-3.7964
-4.6032
-3.7303
-4.5863
-3.6698
-4.5824
-3.6114
-4.6295
-3.5528
-4.6183
-3.5002
-4.7161
95%
Up B.
-4.0072
-4.1679
-4.3009
-4.4202
-4.5296
-4.6345
-4.7335
-4.8296
-4.9237
-5.0119
x=70
PI
Actual value
Low B.
-3.2149
-3.6268
-3.0828
-3.6153
-2.9785
-3.6766
-2.8872
-3.6679
-2.8057
-3.7297
-2.7290
-3.7503
-2.6593
-3.8027
-2.5917
-3.7810
-2.5258
-3.8295
-2.4652
-3.8685
95%
Up B.
-1.8425
-1.9231
-1.9922
-2.0574
-2.1149
-2.1721
-2.2258
-2.2808
-2.3306
-2.3799
x=90
PI
Actual value
Low B.
-1.3199
-1.5932
-1.2565
-1.5937
-1.2027
-1.6224
-1.1537
-1.5202
-1.1115
-1.6183
-1.0694
-1.5957
-1.0328
-1.6630
-0.9931
-1.6424
-0.9589
-1.6481
-0.9258
-1.6944
95%
Up B.
-0.9796
-0.9986
-1.0165
-1.0340
-1.0512
-1.0676
-1.0863
-1.1022
-1.1183
-1.1350
x=100
PI
Actual value
Low B.
-0.6158
-0.7771
-0.6030
-0.7818
-0.5899
-0.7872
-0.5783
-0.7302
-0.5671
-0.7838
-0.5569
-0.7632
-0.5434
-0.8045
-0.5334
-0.7979
-0.5232
-0.7949
-0.5122
-0.8178
95%
Up B.
-2.7732
-2.8888
-2.9855
-3.0754
-3.1572
-3.2372
-3.3123
-3.3865
-3.4590
-3.5266
x=80
PI
Actual value
Low B.
-2.0580
-2.4174
-1.9652
-2.4902
-1.8910
-2.4653
-1.8231
-2.5002
-1.7632
-2.5275
-1.7058
-2.5157
-1.6530
-2.5828
-1.6010
-2.5627
-1.5502
-2.5732
-1.5045
-2.6045
95%
Up B.
-2.9560
-3.0980
-3.2134
-3.3210
-3.4180
-3.5114
-3.5991
-3.6848
-3.7681
-3.8470
x=80
PI
Actual value
Low B.
-2.2346
-2.5579
-2.1186
-2.6364
-2.0281
-2.6282
-1.9464
-2.6194
-1.8737
-2.6573
-1.8063
-2.6116
-1.7439
-2.7440
-1.6836
-2.7161
-1.6255
-2.7657
-1.5721
-2.7853
(d) Spain
t
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
t
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
Table 4: 95% prediction intervals and actual values of the log-central death rates
13
Figure 1: CV and RMSE Histograms
(b) RMSE Histogram
(a) CV Histogram
Figure 2: Observed and forecasted mortality surfaces
(a) France
(b) Italy
(c) Portugal
(d) Spain
14